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The Numerical Study of Transport Phenomena in Porous Media

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The Numerical Study of Transport Phenomena in Porous Media A Numerical Study of Transport Phenomena in Porous Media by May-Fun Liou Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Thesis Adviser: Dr. Isaac Greber Department of Mechanical and Aerospace Engineering Case Western Reserve University August, 2005 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of ______________________________________________________ candidate for the Ph.D. degree *. (signed)_______________________________________________ (chair of the committee) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein. For my beloved parents, Bow-Churng & Chung Wan ii Table of Contents List of Tables vii List of Figures viii Nomenclature xvi Abstract xx 1 Introduction 1 1.1 General Overview 2 1.2 Volume Averaging Method 8 1.3 Three Dimensional Navier-Stokes Equations 13 1.3.1 Fluid-phase Equations 13 1.3.2 Solid-phase Equation 15 1.4 Material under Studied 15 1.5 Outline of the Thesis 16 2 An Enabling Technique for Pore-scale Modeling 18 2.1 Introduction 18 2.2 Mesh-based Microstructure Representing Algorithm 22 2.2.1 Mesh Generator 28 2.2.2 Random Number Generator 32 2.2.3 Quality Control 34 2.2.4 Porosity 37 2.3 Formation of the Two - dimensional Geometry 38 iii 2.4 Formation of the Three - dimensional Geometry 40 2.5 Numerical Solutions 43 2.5.1 Governing equations - both fluid and solid phases 43 2.5.2 Boundary conditions 44 2.5.3 Numerical methods 46 2.6 Sensitivity Test 47 2.6.1 Geometric types 47 2.6.2 Grid dependence study 47 2.6.3 Convergence study in terms of residual 49 2.7 Discussion 50 2.8 Summary 51 3 Fluid Flow and Pressure Drop in Porous Media 53 3.1 Introduction 53 3.2 Validation Cases 56 3.2.1 Laminar flow over a square cylinder 56 3.2.2 Laminar flow over a circular cylinder 57 3.3 Numerical Experiments of 2D Porous Flow 60 3.4 Results and Discussion 61 3.4.1 Two-dimensional channel flow 62 3.4.1.1 Velocity field 63 3.4.1.2 Pressure drop and porosity 70 3.5 Summary 73 iv 4 Forced Convection Heat Transfer in Porous Media 74 4.1 Introduction 74 4.2 Validation Cases – forced convection in single plate-plate channel 84 4.2.1 Bottom plate as an inner plate 86 4.2.2 Bottom plate as an external plate 90 4.3 Numerical Setup for Forced Convective Heat Transfer 94 4.3.1 Two-dimensional channel flow 95 4.3.1.1 Results and discussion 97 4.3.1.2 Effective thermal conductivity 117 4.3.1.3 Thermal dispersion 122 4.3.2 Summary 137 4.4 Three-dimensional Rectangular Duct Flow 138 4.4.1 Flow with Heated Porous Section 138 4.4.2 Summary 150 4.5 Conclusion 150 5 Closure 5.1 Summary 152 5.2 Future Work 154 Appendix A Programs of Random number generating 156 v Appendix B Governing equations and numerical methods 160 Governing equations 160 Numerical method 162 Appendix C Fluid flow and pressure drop data 168 References 169 vi List of Tables Table 1 : List of thermophysical properties of solid materials under study……… 16 Table 2 : Results from grid-dependence study………………………………….. 48 Table 3 : Summary of previous works in evaluating viscous resistance (pressure drop) in porous media……………………………………………………….. 54 Table 4 : List of pressure drops for various porous systems…………………….. 61 Table 5 : Summary of numerical simulations of forced convective heat transfer in Porous media………………………………………………………….. 80 Table 6 : The effective heat conductivity of fluid phase at Re = 270…………… 120 Table 7 : The ratio of effective heat conductivity of fluid phase, f, at Re = 1,100. 121 Table 8 : The ratio of effective heat conductivity of fluid phase, f, at Re = 2,200.. 122 Table C-1: The pressure drop of a simplified electronic cooling system at various conditions………………………………………………………………. 168 vii List of Figures 1.1 Schematic of porous medium, flow variations in the representative Elementary Volume and the pore velocity vector Vp....................................................... 8 2.1 A Schematic explaining the procedures of generating a numerical porous sample………………………..................................................................... 28 2.2 A porous sample with non-uniform porosity……………………………… 34 2.3 Schematic of typical domain of a porous medium – two phase problem…. 37 2.4 Randomly formed solid matrix representing porous medium in 2-D domain (a) fibrous shape formed with triangular mesh, (b) small solids in circular shape formed with triangular mesh, (c) large solids in circular shape with hybrid mesh……………………… 39 2.5 Open-cell foam microstructure …………………………………………… 41 2.6 (a) the cross-section of a typical mesh-based porous medium is shown with solid balls in black and space taken by fluid in white……………. 41 2.6 (b) The cross-section of numerical generated open-cellular sample after converting all solid and fluid tags from Figure 2.8(a), into the oppose ones……………………………………………………………………... 42 2.6 (c) 3D Numerical fluid and solid mesh in cross cut planes along the axial direction – x direction. …………………..……………………………… 42 viii 2.7 Convergence history of residuals from solving Navier-Stokes equations for fluid and solving heat conduction equations for solid in the case of the channel flow heated from both top and bottom walls. …………………...………… 50 3.1 Streamlines of a low Reynolds number flow exhibits symmetry around the solid block ………………………………………………………………… 57 3.2 A hybrid mesh is used for studying unsteady laminar flow over a stationary circular cylinder. ………...………………………………………………… 58 3.3 Contours of u velocity indicate that the laminar flow is fully developed in the flow field …..……………………………………………………………… 59 3.4 Schematic numerical setup for measuring pressure drop …………………... 60 3.5(a) The velocity distribution along the distance from the bottom wall, y', with respect to sampling locations of ∆1, ∆2, ∆3, ∆4, ∆5 and at the exit of the channel …...………………………………………………………………. 64 3.5(b) The velocity distributions in the axial (X) and vertical direction (Y) respectively ..………… ..………………… ……………………………… 65 3.6(a) Velocity contours of the whole numerical domain for the inflow velocity 0.1m/s ………………………..……………………………………….... 66 3.6(b) Vortices contours of the whole numerical domain for the inflow velocity 0.1m/s …………………………………………………………………… 67 3.7 (a) The pressure distribution along the non-dimensional axial direction with respect to several y locations close to the wall……………………. …… 68 3.7 (b) The pressure distribution along the non-dimensional axial direction at three y locations, farther away from the wall………………………... 69 ix ∗ 3.8 Dimensionless pressure gradient parameter, P vs. Red, of various porous systems in Table 3 ………………………………………………………….. 70 ∗ 3.9 Non-dimensional pressure drop P vs. Red in a wide range of Reynolds numbers for two values of porosities.……………………………………… 71 ∗ 3.10 A close-up view of the non-dimensional pressure drop P vs. Red, exhibiting different behaviors caused by varying porosityε …………………………. 72 4.1 Schematic cooling of electronic board with a substrate between coolant and board. ………………………………………………………………………. 85 4.2 Silicon rubber substrate with isothermal walls at 900 K, (a) velocity contours, (b) temperature contours…………………………………….……………… 87 4.3 Local Nusselt number Nux versus dimensionless x of heat transfer at the contact surface of coolant and substrate of silicon rubber………………………. 88 4.4 Aluminum substrate with constant wall temperature, Tw=900K, (a) velocity contours, (b) temperature contours……………………………………… 89 4.5 Silicon rubber substrate with Tw = 900 K and the other parallel plate insulated (a) velocity contours, (b) temperature contours………………………… 90 4.6 Local Nusselt number Nux versus dimensionless x of heat transfer at the contact surface of coolant and substrate of aluminum………………… 92 4.7 Aluminum substrate with isothermal top wall at Tw = 900 K, and the lower parallel plate insulated: (a) velocity contours, (b) temperature contours. 93 4.8 Nux along longitudinal direction x at the contact surface of coolant and substrates of silicon rubber and aluminum…………………………….. 94 x 4.9 Schematic of a two-dimensional porous channel with heat source from below or above…………………………………………………………. 95 4.10(a) Temperature contours of the nonporous channel with the background showing the mesh used………………………………………………… 99 4.10(b) The invariant temperature profiles along the transverse direction in the channel at various sampling positions along the axial direction………. 99 4.10(c) Velocity profiles along the transverse direction at all sampling positions.100 4.10(d) Local Nusselt number Nux versus the longitudinal coordinate x of the baseline case…………………………………………………………… 100 4.11(a) Temperature contours of the porous sample in a channel, along with the mesh used and the porous region in which the solid matrices are shown in black…………………………………………………………………… 101 4.11(b) The invariant temperature profiles along the direction of y* in the channel are plotted for various sampling positions……………………………. 102 4.11(c) Profiles of non-dimensional x-component velocity along the transverse direction at various sampling locations………………………………… 104 4.11(d) Plot of local Nusselt number Nux versus the longitudinal coordinate x… 104 4.12 The ratio, f, of local heat conductivity of fluid phase to that at the inflow condition (300K and 1atm) in terms of the distance away from wall at various sampling locations……………………………………………… 105 4 .13 Distributions of local dimensionless heat transfer along the bottom wall of four solid phases, the upward spikes indicate the enhancement of heat transfer contributed from the physical blockages of porous media……. 108 xi 4.14 Vorticity distribution along the dimensionless transverse direction, y, on the mid-plane of the porous medium (∆3)…………………………………. 109 4.15 Vorticity distributions on the center plane of the hypothetical solid/air (system of θ = 1.0) under Case (b)……………………………………... 111 4.16 Local dimensionless heat transfer rate along the bottom plate at various Reynolds numbers.
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